How to train dweller and improve their SPECIAL stats

• Strength: Weight Room
• Perception: Armory
• Endurance: Fitness Room
• Charisma: Lounge
• Intelligence: Classroom
• Agility: Athletics Room
• Luck: Game Room

Beyond that you have special new versions of the main three resource rooms, and eventually a Nuka-Cola bottling factory, which confers both food and drink, but shockingly no extra Caps, even in tiny amounts. Training goes faster if more than one person trains at a time. Likewise, training rooms do indeed count towards happiness, as the 'right room assignment' mission objective helped me understand.

Aside from the SPECIAL stats, vault dwellers can also be equipped with an outfit and a weapon. Outfits confer stat bonuses and weapons are used to fight off Radroaches, raiders, or to help your dweller survive if you send them scouting for items and Caps out in the wastes. They also level up over time, but leveling up doesn’t increase their stats. It only increases their health, which is still very important for fighting off those previously mentioned random disasters. Vice Versa, people in training rooms only increase stats, and don't level up.

Fun fact. The game is smart enough to realize that when you pull in people from surrounding rooms to fight a disaster that you want those people BACK in those rooms right after. I’ve almost never had issues with getting people back in their assigned rooms. Also, weapon management is a pain if you give a bunch of people weapons and can't remember who. Then you have to tap on each person and check, one by one. Make sure if you give people weapons for a crisis, you take them back immediately.